Biostatistics: Life in Probabilities Cecilia Cotton Department of - - PowerPoint PPT Presentation

Biostatistics: Life in Probabilities

Cecilia Cotton

Department of Statistics and Actuarial Science For the Love of Math and Computer Science October 14, 2017

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Good Morning!

What is Biostatistics? Using the tools of statistics, biostatisticians help answer pressing research questions in medicine, biology and public health, such as whether a new drug works, what causes cancer and other diseases, and how long a person with a certain illness is likely to survive.1 Today’s Goal: Explore a specific topic in public health using probability theory and statistics to help us gain a deeper understanding of the burden of disease in individuals and the population

1https://www.biostat.washington.edu/about/biostatististics

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World Health Organization - Measles Fact Sheet2

◮ Measles is a highly contagious virus is spread by coughing and sneezing, close

personal contact or direct contact with infected nasal or throat secretions

◮ In 1980, before widespread vaccination, measles caused an estimated 2.6 million

deaths each year

◮ Measles is one of the leading causes of death among young children even though a

safe and cost-effective vaccine is available

◮ In 2015, there were 134 200 measles deaths globally ◮ From 2000-2015, measles vaccination prevented an estimated 20.3 million deaths ◮ In 2016, about 85% of the world’s children received one dose of measles vaccine

by their first birthday through routine health services

◮ No specific antiviral treatment exists for the measles virus

2http://www.who.int/mediacentre/factsheets/fs286/en/

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Measles in Canada

◮ Measles was eliminated from Canada in 1998

◮ “The absence of endemic measles transmission in a defined geographic area for 12

months or more, in the presence of a well-performing surveillance system”3

◮ Number of measles cases in Canada (Public Health Agency of Canada)4

◮ 2014: 127 cases ◮ 2015: 196 cases ◮ 2016: 11 cases ◮ 2017: 45 cases (up to September 23, 2017)

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3http://www.wpro.who.int/immunization/documents/measles elimination verification guidelines 2013/en/ 4https://www.canada.ca/en/public-health/services/diseases/measles/surveillance-measles.html 5http://www.cbc.ca/news/canada/toronto/toronto-measles-exposure-1.4263354

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Measles surveillance in Canada: 20156

◮ 196 cases in four provinces, 5.5 cases per 1,000,000 population

◮ 172 cases were never immunized or immunization was not up-to-date for age ◮ 9 cases were age-ineligible for measles-containing vaccine ◮ 7 cases were up-to-date with measles vaccination ◮ Remaining cases had unknown vaccination status

◮ 9 imported cases originating from China (2), India (2), Ethiopia, Pakistan, South

Africa, Tunisia, and United States

◮ 4 outbreaks (190 cases) and 6 spontaneous sporadic cases ◮ Most cases (81.1%, n=159) were in a non-immunizing religious community. Index

case was exposed to measles during travel to “a popular theme park in California”

6Sherrard L, Hiebert J, Cunliffe J, Mendoza L, Cutler J. Measles surveillance in Canada: 2015.

Can Comm Dis Rep 2016;42(7):139-145 5/30

Vaccination and Herd Immunity

◮ Measles can be prevented with a 2 dose vaccine (Ontario: 12 months & 4 years)7 ◮ The direct effect of vaccination is to protect the vaccinated individual from

contracting measles

◮ Herd Immunity is the indirect protection of unvaccinated persons, whereby an

increase in the prevalence of vaccine-immunity prevents circulation of infectious agents in unvaccinated susceptible populations.8

◮ The vaccination rate necessary to achieve herd immunity depends on the vaccine

efficacy, and the infectiousness of the disease

◮ Let’s explore the effect of vaccination rate using some demonstrations

7http://www.health.gov.on.ca/en/pro/programs/immunization/docs/immunization schedule.pdf 8Kim TH, Johnstone J, Loeb M. Vaccine herd effect. Scandinavian Journal of Infectious Diseases.

2011;43(9):683-689. 6/30

Herd Immunity Demonstration

◮ If your handout has an empty box (Round 1 and/or Round 2) answer the

following question:

◮ Does your birthday occur on a day greater than or equal to the 4th of the month?

◮ Birthday: February 3 - write NO in the box ◮ Birthday: May 4 - write YES in the box ◮ Birthday: August 30 - write YES in the box

◮ Now our population is about to receive 5 visitors infected with measles

◮ If the visitor infects their initial contact, everyone is eventually exposed ◮ If the visitor does not infect their initial contact, there are no further exposures

◮ Round 1: 75% vaccination rate; Round 2: 95% vaccination rate ◮ Online simulation from The Guardian9

9https://www.theguardian.com/society/ng-interactive/2015/feb/05/-sp-watch-how-measles-outbreak-spreads-when-kids-get-vaccinated

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https://www.theguardian.com/society/ng-interactive/2015/feb/05/-sp-watch-how-measles-outbreak-spreads-when-kids-get-vaccinated 8/30

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Model Specification and Assumptions

◮ n = 100 individuals in each population

◮ All in close contact and mix regularly ◮ Each equally likely to come into to contact with an outsider

◮ Vaccination Rate = Prob[vaccination] = p (unspecified, for now) ◮ Vaccine Efficacy = Prob[protected | vaccinated] = 0.99 ◮ Each population gets 5 visits from an infectious outsider

◮ If they contact a protected individual: measles does not enter the community ◮ If they contact a susceptible individual (unvaccinated or vaccinated but susceptible):

◮ Risk of Infection = Prob[infection | susceptible] = 0.90 ◮ If measles enters community, all susceptible individuals will eventually be exposed

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Susceptible Unvaccinated Susceptible Uninfected 10% Infected 90% 100% (100 − p) Vaccinated Protected Uninfected 100% 99% Susceptible Uninfected 10% Infected 90% 1% p

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Some Probability Definitions and Rules

◮ The Probability than event A occurs is denoted P(A) where 0 ≤ P(A) ≤ 1 ◮ The probability of an event not occurring is one minus the probability of it

• ccurring: P(¯

A) = 1 − P(A)

◮ If A and B are two events in the sample space S, then the Conditional Probability

• f A given B is:

P(A|B) = P(A ∩ B) P(B) , when P(B) > 0

◮ Law of Total Probability: If B1, B2, . . . is a partition of the sample space S, then

for any event A: P(A) =

• i

P(A ∩ Bi) =

• i

P(A|Bi)P(Bi)

◮ This simplifies to P(A) = P(A|B)P(B) + P(A|¯

B)P(¯ B)

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Probability of Infection

◮ Define the following events for a random individual

◮ V = vaccination ◮ P = protection from measles ◮ I = infection

◮ First we condition on whether or not individual was vaccinated

P(I) = P(I|V )P(V ) + P(I| ¯ V )P( ¯ V )

◮ If the individual is vaccinated we have:

P(I|V )P(V ) = P(I|V ∩ P)P(P|V )P(V ) + P(I|V ∩ ¯ P)P(¯ P|V )P(V ) = 0.90(0.01)p

◮ For the unvaccinated:

P(I| ¯ V )P( ¯ V ) = P(I| ¯ V ∩ P)P(P| ¯ V )P( ¯ V ) + P(I| ¯ V ∩ ¯ P)P(¯ P| ¯ V )P( ¯ V ) = 0.90(1 − p)

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Susceptible Unvaccinated Susceptible Uninfected 0.10 Infected 0.90(1 − p) 0.90 1 (1 − p) Vaccinated Protected Uninfected 1 0.99 Susceptible Uninfected 0.10 Infected 0.90(0.01)p 0.90 0.01 p

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Probability of an “Outbreak” in the Population

◮ We just showed: P(Infection) = 0.90(1 − p) + 0.90(0.01)p = 0.90(1 − 0.99p) ◮ Recall: Each population gets 5 visits from an infectious outsider ◮ What is the probability that an “outbreak” occurs in a given population?

P(Outbreak) = 1 − P(No Outbreak) = 1 −

5

• i=1

P(ith visit does not lead to an infection) = 1 − [P(No Infection)]5 = 1 − [1 − 0.90(1 − 0.99p)]5

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Probability of an “Outbreak” in the Population

Vaccination Theoretical Rate Probability 10.00 99.98 30.00 99.33 50.00 95.17 58.50 90.75 68.90 81.46 74.40 74.16 83.80 56.49 86.00 51.22 90.00 40.33 95.00 24.06 99.70 5.70

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Simulation Study

◮ We will further explore the population dynamics using Monte Carlo Simulation ◮ Frequently used in biostatistics when exact analytical derivations are not possible ◮ A typical Monte Carlo Simulation involves the following:10

• 1. General N independent data sets under the conditions of interest
• 2. Compute the numerical values of the estimator of the statistic of interest T(data)

for each data set: T1, . . . , Tn

• 3. If N is large enough, summary statistics across T1, . . . , Tn should be good

approximations of the true properties of the estimator under the conditions of interest

10http://www4.stat.ncsu.edu/ davidian/st810a/simulation handout.pdf

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Simulation Study

◮ For this study, simulate N = 1000 different populations of size n = 100 ◮ Each individual’s vaccination and protection/susceptibility status were

independently generated from Binomial random variables

◮ Up to 5 visits from an infectious outsider, Infection status also randomly generated ◮ Statistics of interest:

◮ Does an “Outbreak” occur? (i.e. does the infection enter the population?) ◮ If the infection enters the population, what proportion of individuals are: ◮ Unvaccinated, Uninfected ◮ Unvaccinated, Infected ◮ Vaccinated, Protected, (Uninfected) ◮ Vaccinated, Susceptible, Uninfected ◮ Vaccinated, Susceptible, Infected

◮ Repeat the simulation study for 11 different vaccination rates ◮ Coding was done using R statistical programming language

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Simulation Results

Vaccination Infection Theoretical Rate Present Probability 10.00 99.90 99.98 30.00 99.50 99.33 50.00 95.40 95.17 58.50 89.90 90.75 68.90 79.60 81.46 74.40 74.00 74.16 83.80 52.80 56.49 86.00 52.40 51.22 90.00 42.70 40.33 95.00 23.50 24.06 99.70 6.20 5.70

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Simulation Results

When Infection Enters Population Unvaccinated Vaccinated Vaccinated Unvaccinated Vaccinated Vaccination Infection Susceptible Protected Susceptible Rate Present Uninfected Uninfected Uninfected Infected Infected 10.00 99.90 8.82 0.01 9.90 81.17 0.10 30.00 99.50 6.90 0.03 29.72 63.07 0.28 50.00 95.40 4.84 0.05 49.77 44.87 0.46 58.50 89.90 4.15 0.06 57.71 37.55 0.52 68.90 79.60 3.04 0.06 67.91 28.42 0.57 74.40 74.00 2.51 0.07 73.45 23.28 0.68 83.80 52.80 1.51 0.08 82.61 15.02 0.79 86.00 52.40 1.39 0.10 84.66 13.04 0.82 90.00 42.70 0.98 0.10 88.56 9.48 0.89 95.00 23.50 0.49 0.11 93.25 5.20 0.95 99.70 6.20 0.03 0.06 97.63 0.60 1.68

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Simulation Results

When Infection Enters Population Unvaccinated Vaccinated Vaccinated Unvaccinated Vaccinated Vaccination Infection Susceptible Protected Susceptible Rate Present Uninfected Uninfected Uninfected Infected Infected 10.00 99.90 8.82 0.01 9.90 81.17 0.10 30.00 99.50 6.90 0.03 29.72 63.07 0.28 50.00 95.40 4.84 0.05 49.77 44.87 0.46 58.50 89.90 4.15 0.06 57.71 37.55 0.52 68.90 79.60 3.04 0.06 67.91 28.42 0.57 74.40 74.00 2.51 0.07 73.45 23.28 0.68 83.80 52.80 1.51 0.08 82.61 15.02 0.79 86.00 52.40 1.39 0.10 84.66 13.04 0.82 90.00 42.70 0.98 0.10 88.56 9.48 0.89 95.00 23.50 0.49 0.11 93.25 5.20 0.95 99.70 6.20 0.03 0.06 97.63 0.60 1.68

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Simulation Results

When Infection Enters Population Unvaccinated Vaccinated Vaccinated Unvaccinated Vaccinated Vaccination Infection Susceptible Protected Susceptible Rate Present Uninfected Uninfected Uninfected Infected Infected 10.00 99.90 8.82 0.01 9.90 81.17 0.10 30.00 99.50 6.90 0.03 29.72 63.07 0.28 50.00 95.40 4.84 0.05 49.77 44.87 0.46 58.50 89.90 4.15 0.06 57.71 37.55 0.52 68.90 79.60 3.04 0.06 67.91 28.42 0.57 74.40 74.00 2.51 0.07 73.45 23.28 0.68 83.80 52.80 1.51 0.08 82.61 15.02 0.79 86.00 52.40 1.39 0.10 84.66 13.04 0.82 90.00 42.70 0.98 0.10 88.56 9.48 0.89 95.00 23.50 0.49 0.11 93.25 5.20 0.95 99.70 6.20 0.03 0.06 97.63 0.60 1.68

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Simulation Results

When Infection Enters Population Unvaccinated Vaccinated Vaccinated Unvaccinated Vaccinated Vaccination Infection Susceptible Protected Susceptible Rate Present Uninfected Uninfected Uninfected Infected Infected 10.00 99.90 8.82 0.01 9.90 81.17 0.10 30.00 99.50 6.90 0.03 29.72 63.07 0.28 50.00 95.40 4.84 0.05 49.77 44.87 0.46 58.50 89.90 4.15 0.06 57.71 37.55 0.52 68.90 79.60 3.04 0.06 67.91 28.42 0.57 74.40 74.00 2.51 0.07 73.45 23.28 0.68 83.80 52.80 1.51 0.08 82.61 15.02 0.79 86.00 52.40 1.39 0.10 84.66 13.04 0.82 90.00 42.70 0.98 0.10 88.56 9.48 0.89 95.00 23.50 0.49 0.11 93.25 5.20 0.95 99.70 6.20 0.03 0.06 97.63 0.60 1.68

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Results and Limitations

◮ Herd immunity protects individuals in the population who can/have not be

immunized or who are unprotected by immunization

◮ A high vaccination rate is necessary to achieve herd immunity ◮ Even at a vaccination rate of 95% outbreaks will still occur ◮ We made a number of simplifying assumptions today which could be relaxed: ◮ Not all members of the population randomly mix with each other ◮ Did not explore the effects of Vaccine Efficacy or Risk of Infection

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Ontario Perspectives

◮ Public Health Ontario: Immunization coverage report for school pupils 11 ◮ Up-to-date coverage refers to the proportion of a population that has received the

recommended number of doses of a certain vaccine by a certain age. Many children who are not up-to-date have received some, but not all, recommended doses in a vaccine series

◮ Summary of Key Findings

◮ Immunization coverage estimates vary greatly in Ontario based on the vaccine

program, the age group assessed and by geographic region

◮ With some exceptions, Ontario falls short of most immunization coverage goals ◮ Immunization coverage may be underestimated if students have been immunized and

the information has not been captured by the immunization surveillance system

11Ontario Agency for Health Protection and Promotion (Public Health Ontario). Immunization coverage

report for school pupils: 2013-14, 2014-15 and 2015-16 school years. Toronto, ON: Queen’s Printer for Ontario; 2017. 25/30

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Discussion

◮ These sorts of investigations and simulation studies are important to public health ◮ Simulation results can complement and extend real-world observational data ◮ Studies can provide insight as to where to focus limited public health resources

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Thank you! Have a great day!

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